Chapter 4.1 - Integers and the Number line

In the previous sections we saw some situations where negative numbers come into play. In this section we will see some more examples. 

The fig.4.3 below  shows a multi storey building. It has 3 floors below ground level and several floors above ground level. 

Fig.4.3

• The floor which is exactly at the ground level is named as 'floor 0'. 
• The floor just above it is 'floor 1'. 
• The next upper floor is 'floor 2'. 
• Similarly, the floor just below floor 0 is 'floor -1'. 
• The one just below floor -1 is 'floor -2 ... and so on. 
• This procedure of naming the floors is used in the lifts also. 
      ♦ When the lift is in the lower most floor of the building, the indicator in the lift shows '-3'. 
      ♦ When it climbs one floor, the indicator shows '-2'...and so on. So higher the negative number, lower is the lift 
    ♦ When the lift reaches the ground floor, '0' is shown.

Another situation is the expression of temperature. During cold weather the temperature begins to decrease. When the temperature decreases further, water begins to solidify. The exact temperature at which the water solidifies and becomes ice, is taken as 0^o centigrade. But the temperature can decrease even further. It is important to know how much below zero is the temperature in a locality. For example, this knowledge is important in the design of heating systems. A heating system which is capable to provide normal living conditions at zero degree centigrade will not be sufficient if the temperature is much lower than zero. Measurement of the exact temperature below zero is of vital importance in many Scientific and Engineering experiments and procedures.  

So we indeed need a method to measure temperatures lower than zero. Negative numbers are an easy method to do it. If the whether department says that the temperature is -20^o centigrade, it means, the temperature is 20 degrees below zero. If it is -25^o, it is even colder than -20^o by 5 degrees. So higher the negative number, colder is the temperature.  

Thus we find that negative numbers are an important part of our life. We must include them in the family of other numbers. Let us see how this is done:

• Numbers 1, 2, 3, . . ., are the Natural numbers.
• If we add '0' to this, the list becomes 0, 1, 2, 3, . . . The name of this list is Whole numbers
• Now we just met negative numbers. We must add them also to the list of whole numbers. Then we get: . . . -5, -4, -3, -2, -1, 0, 1, 2, 3, . . ., This list is known as the list of Integers.

On a number line, the integers can be given appropriate places as shown in fig.4.4 below. The arrows at the ends indicate that the line can extend to any distance on either side. Thus it can accommodate up to any large number, be it positive or negative. Also the spacing between the numbers should be uniform. This is a rule which have to be strictly followed while drawing number lines.

Fig.4.4 The number line

Let us see how the above number line works. Suppose we are standing at +5. If we begin to move towards the left, the value begins to decrease. It becomes +4, then +3, then +2 and so on. When we reach zero, there is no value. If we continue moving towards the left, the value will fall even below zero. It becomes -1, then -2, then -3 and so on. Similarly, from any where on the number line, if we move towards the right, the value increases.

• Zero can be taken as a 'bench mark'. 
• The further left we are from zero, the lesser is our value. Also, as we are on the left of zero, the value will be negative
• The further right we are from zero, the greater is our value. Also, as we are on the right of zero, the value will be positive

Thus, the number line shows the position of each integer in the real world. Take for example -6 and -5. The 'numerical value' of -6 is greater than -5 (because, if we remove the '-' sign, 6 is greater than 5). But when the '-' sign is attached, things are just the opposite. -6 has a lower value than -5.

We can think of it in this way:
There is a greater distance to travel from -6 than from -5 to get to the 'boundary zero' and then to the positive side. So any body would prefer -5. So, as we move further towards the left, the value gets lower and lower.

Based on the above discussion, we can now see some solved examples:
Solved example 4.1
In the fig.4.5 given below, fill up the boxes using ‘<’ OR ‘>’

Fig.4.5

Solution:
(a) -20 > -35 (b) -9 < 2 (c) 60 > -62 (d) 0 > -1 (e) -70 > -71 (f) -73 < -7
Solved example 4.2
Mark the following points on a number line: (a) -7 (b) -1 (c) +4 (d) +8 (e) -4
Solution:
The fig.4.6 shows the required markings

Fig.4.6

Solved example 4.3
Given below are a few pairs of numbers.
(a) 4,12 (b) -21, -22 (c) 1, -100, (d) -5, -11
In each pair, pick out the number which is to the right of the other on the number line
Solution:
(a) 12. It is to the right of 4
(b) -21. It is to the right of -22
(c) 1. It is to the right of -100
(d) -5. It is to the right of -11
Solved example 4.4
Write all the integers between 4 and -7 in the increasing order of value.
Solution:
-6, -5, -4, -3, -2, -1, 0, 1, 2 and 3
Solved example 4.5
If we move from -20 to -10, will the values of the integers increase or decrease?
Solution:
Values will increase.

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So now we have a good understanding about the positive and negative numbers, and their positions in the real world. In the next section we will see how addition between these numbers can be carried out.

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Chapter 4 - Integers - Negative numbers

In the previous sections we completed the discussion on direct and inverse proportions. In this chapter, we discuss a more basic topic of integers. Consider the following situation:

A chef is making fruit salad for a party. He wants 16 apples. But in the kitchen there are only 12. He needs 4 more. He goes to a nearby home and borrow 4 apples. Now he has all the required 16 apples. But he is in debt. He has to return the 4 apples to the nearby home. So he makes a note of it. In his diary he writes: 'Number of apples = 4'.

But writing simply like this may cause problems. After some time, looking at the diary, he may wonder what the number 4 really indicate?. Does he want 4 apples for himself, or does he have to give 4 apples to some one else?

So he must write it more clearly:
• No. of apples to be given away = 4. The mathematical way of writing this is:
• No. of apples = - 4. 
In the second method, '-' (minus) sign is put before the number 4. This '-' sign has much significance. When the new stock of apples comes to the kitchen, the chef has to take 4 from it and give it to the nearby home. If this new stock has say, 25 apples, the chef will be left with 21 apples (since 25 – 4 = 21) only.

Another example:
The fig.4.1 below shows the diagram of a large water tank in a town. During the day time, people use water for various activities like bathing, cooking, washing etc., So the water level decreases. But during the night, there are no such activities, and so there is no decrease in water level during the night. And also, during the night, the used up water is replenished by water from a nearby treatment plant. So the water level rises to the previous level.

Fig.4.1

Thus the water level in the tank always maintains a constant level each morning. We can see from the fig. 4.1 that, this level is indicated as 'zero' in the scale.

But during summer, there will be scarcity of water. All the water that is used during day cannot be replenished during the night. So the water level will not rise back to zero each morning. It will always be below zero. At mid summer, when the intensity of heat is high, the level will go further and further below zero. The care taker of the water tank has to keep a close watch. He must tell others about the level of water in the tank every morning during summer. How would he say it?   

He checks and finds by how many units the level is below zero. If it is 3, he would say: “The level is 3 units below zero”. An easier way is to say mathematically : “ The level is -3”. But for this, a simple modification has to be made to the scale in the tank. That is., a '-' sign should be placed in front of all the units below zero. This is shown in the fig.4.2 below.

Fig.4.2

If the level is 5 units below zero, he can say: “Level is -5”. The '-' sign instantly gives a clear picture of the seriousness of the situation.

Numbers with '-' sign in front are called negative numbers. All  the readings below zero are negative readings.

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We must note one peculiarity about these negative readings: As the level decreases, the negative value increases. For example, if the level falls even below -5 and reach -7, it means that we have still lesser water to use. But the value after the '-' sign has increased from 5 to a larger value of 7.

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There are many situations where negative numbers are used in day to day life. In the next section we will see some more examples.

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